The effort is a direct challenge to another branch of the composer’s family, which is also lobbying to run the hallowed precincts, where Wagner’s operas are reverentially presented each summer.

“My vision is a Bayreuth of the 21st century, a new artistic spirit,” Ms. Wagner, 63, said in a telephone interview from Weimar, Germany, where she runs her own arts festival. “We are not traditionalists, even though we have lots of respect for tradition,” she added. “We think Wagner was a composer of the future, and we always respect this spirit.”

Ms. Wagner said that if their bid succeeded, the festival and the City Opera would work together in some way, possibly by collaborating on productions. They were also proposing a separate miniseason at the festival’s theater in June, perhaps a long weekend in which the works of other composers would be performed, Ms. Wagner said. She called it a “kind of laboratory.”

Mr. Mortier did not respond to telephone messages but sent an e-mail statement through his spokesman, Pascal Nadon, at City Opera. Mr. Mortier said that he did not expect the festival’s six-week season to interfere with his job in New York, which he assumes next fall. “On the contrary, it could reinforce the international stature of City Opera,” he said. Mr. Nadon said that the collaboration could lead to joint productions.

Mr. Mortier and Ms. Wagner plan to present a seven-page proposal to the festival’s governing foundation on Monday. The festival would not comment on the development. An assistant to Peter Emmerich, the festival’s spokesman, said that Mr. Emmerich had declined to come to the telephone.

The Wagner-Mortier proposal was reported on Tuesday in the German press, and The Associated Press said that the festival confirmed receiving an application. A spokesman for the German chancellor, Angela Merkel, said the foundation — which consists of federal, regional and local officials — would weigh the two proposals, Bloomberg News reported.

“It is reasonable to assume that the foundation will look at both concepts and will decide for the most convincing one,” said the spokesman, Thomas Steg. The foundation owns the theater, called the Festspielhaus, and subsidizes the festival.

Wagner himself founded the festival in 1876. His grandsons Wolfgang and Wieland assumed command in 1951; after Wieland Wagner’s death in 1966, his younger brother, Wolfgang, became director for life. The festival is facing deficits. Wolfgang, who turns 89 on Saturday and is ailing, submitted his resignation in April, to take effect after this summer’s festival, which ends on Thursday.

He said that he would resign on condition that his daughter Katharina Wagner, 30, succeeded him. She and a half-sister by a previous Wolfgang marriage, Eva Wagner-Pasquier, have submitted a proposal to run the festival jointly.

Both have opera experience. Ms. Wagner-Pasquier, 63, serves as artistic adviser to the Aix-en-Provence opera festival in southern France and Katharina Wagner, an opera director, presented her first Wagner work in Bayreuth last summer — a roundly booed and critically panned “Die Meistersinger von Nürnberg.”

With the days counting down before the board meets to decide on the succession, Nike Wagner joined forces with Mr. Mortier. She is the daughter of Wieland Wagner, whom many feel provided strong artistic leadership. Mr. Mortier, in his e-mail statement, called Nike Wagner the “favorite of the music press.”

Nike Wagner, a musicologist by training, hinted at the family rivalry in the matter of succession. “I think that Wieland Wagner’s offspring should never have been excluded, or other offspring,” she said. “The family consists of more tribes than just Wolfgang Wagner’s tribe.” She has been critical of Katharina Wagner’s work in the past, calling it “old wine in a new wineskin.”

Nike Wagner said in the end it was more important for Bayreuth to have good leadership than a leader named Wagner. “It’s part of my heritage,” she said. “I’m concerned about how will Bayreuth continue in the future, and I’m engaged in this problem, and I want the best possible leadership. This has priority. And then come the tribe and the Wagner business.”

Nike Wagner has known Mr. Mortier for many years, she said. “I’ve followed him all the way through his artistic activities,” she added. “I admire his work very much. Intellectually our minds are very similar.” Asked how, she said, “In the way of being more future-minded, looking out for new and best ways to present opera, to present the arts.”

Mr. Mortier, who once ran the Théâtre Royale de la Monnaie in Brussels and the Salzburg Festival in Austria, has a reputation of mounting provocative and scandal-tinged productions. He is ending his tenure as director of the Paris National Opera this season.

In his e-mail message, he said, “Part of the Wagner family insisted that I should assist Nike Wagner.” In another message, he said he was concerned about the future of the Bayreuth Festival. At the same time, he added that he wanted to send a signal to City Opera board members who are unhappy with his programming choices and who have not increased financial support. All additional money the opera has raised to put on its season has come from foundations, he said.

Money is also being raised for the renovation of City Opera’s home, the New York State Theater, which is closed this season. The company is putting on only a handful of concert performances while seeking to hold onto its subscribers and patrons.

The Bayreuth Festival has tried a change of leadership once before, in 2001, when Wolfgang Wagner indicated he was ready to retire. The foundation rejected both his choice — his wife, Gudrun, who died last year — and Nike Wagner, and chose Ms. Wagner-Pasquier as director. Wolfgang Wagner then said he would stay in the job, and designated Katharina Wagner as his successor.